Passive noise cancellation means that headphone/earphone works like hearing protector or earplug, it is closed design that isolates your ears (or ear canals) from outside world. Effective and doesnt affect soundquality. IEM (in ear monitors) are most effective in isolating and are small, they are like earplugs. But they cannot give same soundquality what similarly priced fullsize can can give, but that doesnt mean they are bad! Stuffing them to your ears and getting proper seal is bit tricky sometimes. (and proper seal is a MUST to get optimal sound quality)
Downside of passive noise cancelling is that while outside ambience noise is heavily reduced, you start to hear your body noises much more clearly. Walking, cable scratching against your shirt, even heartbeat, they are all audible. Not much of an issue when music playing though.
Active noise cancellation works by adding counter-noise to block outside noise. Of course, this does affect soundquality, but how much depends on headphone and system how their active noise cancelling works. Doesnt seal, but atleast lets you hear music without blasting your ears, like with open non-isolating and non-cancelling headphones in noisy enviroment.